[Author]
Oleg I. Reinov
[Title]
Around Grothendieck's theorem on operators with nuclear adjoins
[AMS Subj-class]
46B28 Spaces of operators; tensor products; approximation properties
[Abstract]
In 1955, A. Grothendieck proved that if an operator $T: X\to Y$ in Banach
spaces has a nuclear adjoint, then $T$ is nuclear provided that $X^*$ has
the approximation property. It was shown by T. Figiel and W.\,B. Johnson (1973)
that the assumption on $X^*$ is essential. A generalization of Grothendieck's
theorem was obtained in 1987 by E. Oja and O. Reinov: The conclusion of the
theorem is true if $Y^{***}$ has the approximation property (and the assumption
posed on $Y^{***}$ is also essential). A "revised" (more general) version of
the theorem was formulated and proved in 2012 by E. Oja for weak${}^*$
continuous operators from a dual Banach space into an arbitrary Banach space.
She considered even the cases of $\alpha$-nuclear operators for tensor norms
$\alpha.$ We present here, among other results, a "revised" version of
Grothendieck's theorem in the cases of $\alpha$-nuclear operators for
projective tensor quasi-norms $\alpha.$ This is a talk at the 27th
St.Petersburg Summer Meeting in Mathematical Analysis (2018) and also
a preprint of the paper that will be submitted to the Journal of Mathematical
Analysis and Applications.
[Comments]
LaTeX, English, 24 pp.
[Contact e-mail]
orein51@mail.ru